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SeqCalc.agda
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SeqCalc.agda
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module SeqCalc where
open import Prop
open import Ctx
open import NatDed hiding (struct)
open import Data.Product
using (∃; ∃-syntax; proj₂; -,_) renaming (_,_ to infix 4 ⟨_,_⟩)
open import Data.Empty using (⊥)
infix 20 _⇒_^_
infixr 21 _+_
data Size : Set where
zero : Size
suc : Size → Size
_+_ : Size → Size → Size
private
variable
P : `Atom
A B C D : `Prop
Γ Δ : Ctx
m n : Size
-- to facilitate termination checking, each sequent is indexed by the
-- size of the derivation
data _⇒_^_ : Ctx → `Prop → Size → Set where
idᵖ : Γ ∋ ` P
→ Γ ⇒ ` P ^ zero
-- this rule only allows atomic proposition to be concluded this
-- matches well with the verificationist point of view the general
-- version for all proposition is admissible (see "id" below)
∧R : Γ ⇒ A ^ m
→ Γ ⇒ B ^ n
→ Γ ⇒ A `∧ B ^ m + n
∧L₁ : Γ ∋ A `∧ B
→ Γ , A ⇒ C ^ m
→ Γ ⇒ C ^ suc m
∧L₂ : Γ ∋ A `∧ B
→ Γ , B ⇒ C ^ m
→ Γ ⇒ C ^ suc m
⊃R : Γ , A ⇒ B ^ m
→ Γ ⇒ A `⊃ B ^ suc m
⊃L : Γ ∋ A `⊃ B
→ Γ ⇒ A ^ m
→ Γ , B ⇒ C ^ n
→ Γ ⇒ C ^ m + n
∨R₁ : Γ ⇒ A ^ m
→ Γ ⇒ A `∨ B ^ suc m
∨R₂ : Γ ⇒ B ^ m
→ Γ ⇒ A `∨ B ^ suc m
∨L : Γ ∋ A `∨ B
→ Γ , A ⇒ C ^ m
→ Γ , B ⇒ C ^ n
→ Γ ⇒ C ^ m + n
⊤R : Γ ⇒ `⊤ ^ zero
⊥L : Γ ∋ `⊥
→ Γ ⇒ C ^ zero
-- structural rules
struct : Γ ⊆ Δ
→ Γ ⇒ C ^ m
→ Δ ⇒ C ^ m
struct Γ⊆Δ (idᵖ a) = idᵖ (Γ⊆Δ a)
struct Γ⊆Δ (∧R a b) = ∧R (struct Γ⊆Δ a) (struct Γ⊆Δ b)
struct Γ⊆Δ (∧L₁ a b) = ∧L₁ (Γ⊆Δ a) (struct (⊆-step Γ⊆Δ) b)
struct Γ⊆Δ (∧L₂ a b) = ∧L₂ (Γ⊆Δ a) (struct (⊆-step Γ⊆Δ) b)
struct Γ⊆Δ (⊃R a) = ⊃R (struct (⊆-step Γ⊆Δ) a)
struct Γ⊆Δ (⊃L a b c) = ⊃L (Γ⊆Δ a) (struct Γ⊆Δ b) (struct (⊆-step Γ⊆Δ) c)
struct Γ⊆Δ (∨R₁ a) = ∨R₁ (struct Γ⊆Δ a)
struct Γ⊆Δ (∨R₂ a) = ∨R₂ (struct Γ⊆Δ a)
struct Γ⊆Δ (∨L a b c) = ∨L (Γ⊆Δ a) (struct (⊆-step Γ⊆Δ) b) (struct (⊆-step Γ⊆Δ) c)
struct Γ⊆Δ ⊤R = ⊤R
struct Γ⊆Δ (⊥L a) = ⊥L (Γ⊆Δ a)
wk : Γ ⇒ C ^ m
→ Γ , A ⇒ C ^ m
wk x = struct S x
wk′ : Γ , A ⇒ C ^ m
→ Γ , B , A ⇒ C ^ m
wk′ x = struct (λ { Z → Z ; (S i) → S (S i) }) x
exch : Γ , A , B ⇒ C ^ m
→ Γ , B , A ⇒ C ^ m
exch x = struct ((λ { Z → S Z ; (S Z) → Z ; (S (S i)) → S (S i) })) x
-- we write ∃[ q ] ⇒ Γ ⇒ A ^ q for sequent with size we don't care,
-- the monadic binding makes manipulating them easier, though I'm not
-- sure whether they actually form a monad
_>>=_ : ∃[ q ] Γ ⇒ A ^ q
→ (∀ {m} → Γ ⇒ A ^ m → ∃[ p ] Δ ⇒ B ^ p)
→ ∃[ r ] Δ ⇒ B ^ r
⟨ _ , x ⟩ >>= f = let ⟨ _ , a ⟩ = f x
in -, a
-- identity theorem
-- this shows global completeness of the calculus
-- which means eliminations (left rules) are strong enough to
-- extract all the information introductions (right rule) put into
id : Γ ∋ A
→ ∃[ q ] Γ ⇒ A ^ q
id {A = ` P} x = -, idᵖ x
id {A = A `∧ B} x = do a ← id Z
b ← id Z
-, ∧R (∧L₁ x a) (∧L₂ x b)
id {A = A `⊃ B} x = do a ← id Z
b ← id Z
-, ⊃R (⊃L (S x) a b)
id {A = A `∨ B} x = do a ← id Z
b ← id Z
-, ∨L x (∨R₁ a) (∨R₂ b)
id {A = `⊤} x = -, ⊤R
id {A = `⊥} x = -, ⊥L x
-- cut theorem
cut : Γ ⇒ D ^ m
→ Γ , D ⇒ C ^ n
→ ∃[ q ] Γ ⇒ C ^ q
-- idᵖ + arbitrary rule
cut (idᵖ a) b = -, struct (λ { Z → a ; (S i) → i }) b
-- arbitrary rule + idᵖ
cut a (idᵖ Z) = -, a
cut a (idᵖ (S b)) = -, idᵖ b
-- right rule + arbitrary rule
cut (∧L₁ a b) c = do x ← cut b (wk′ c)
-, ∧L₁ a x
cut (∧L₂ a b) c = do x ← cut b (wk′ c)
-, ∧L₂ a x
cut (⊃L a b c) d = do x ← cut c (wk′ d)
-, ⊃L a b x
cut (∨L a b c) d = do x ← cut b (wk′ d)
y ← cut c (wk′ d)
-, ∨L a x y
cut (⊥L a) b = -, ⊥L a
-- arbitrary rule + left rule
cut a (∧R b c) = do x ← cut a b
y ← cut a c
-, ∧R x y
cut a (⊃R b) = do x ← cut (wk a) (exch b)
-, ⊃R x
cut a (∨R₁ b) = do x ← cut a b
-, ∨R₁ x
cut a (∨R₂ b) = do x ← cut a b
-, ∨R₂ x
cut a ⊤R = -, ⊤R
-- right rule + left rule
-- the cut proposition is not used
cut o@(∧R a b) (∧L₁ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₁ c x
cut o@(⊃R a) (∧L₁ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₁ c x
cut o@(∨R₁ a) (∧L₁ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₁ c x
cut o@(∨R₂ a) (∧L₁ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₁ c x
cut o@⊤R (∧L₁ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₁ c x
cut o@(∧R a b) (∧L₂ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₂ c x
cut o@(⊃R a) (∧L₂ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₂ c x
cut o@(∨R₁ a) (∧L₂ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₂ c x
cut o@(∨R₂ a) (∧L₂ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₂ c x
cut o@⊤R (∧L₂ (S c) d) = do x ← cut (wk o) (exch d)
-, ∧L₂ c x
cut o@(∧R a b) (⊃L (S c) d e) = do x ← cut o d
y ← cut (wk o) (exch e)
-, ⊃L c x y
cut o@(⊃R a) (⊃L (S c) d e) = do x ← cut o d
y ← cut (wk o) (exch e)
-, ⊃L c x y
cut o@(∨R₁ a) (⊃L (S c) d e) = do x ← cut o d
y ← cut (wk o) (exch e)
-, ⊃L c x y
cut o@(∨R₂ a) (⊃L (S c) d e) = do x ← cut o d
y ← cut (wk o) (exch e)
-, ⊃L c x y
cut o@⊤R (⊃L (S c) d e) = do x ← cut o d
y ← cut (wk o) (exch e)
-, ⊃L c x y
cut o@(∧R a b) (∨L (S c) d e) = do x ← cut (wk o) (exch d)
y ← cut (wk o) (exch e)
-, ∨L c x y
cut o@(⊃R a) (∨L (S c) d e) = do x ← cut (wk o) (exch d)
y ← cut (wk o) (exch e)
-, ∨L c x y
cut o@(∨R₁ a) (∨L (S c) d e) = do x ← cut (wk o) (exch d)
y ← cut (wk o) (exch e)
-, ∨L c x y
cut o@(∨R₂ a) (∨L (S c) d e) = do x ← cut (wk o) (exch d)
y ← cut (wk o) (exch e)
-, ∨L c x y
cut o@⊤R (∨L (S c) d e) = do x ← cut (wk o) (exch d)
y ← cut (wk o) (exch e)
-, ∨L c x y
cut o@(∧R a b) (⊥L (S c)) = -, ⊥L c
cut o@(⊃R a) (⊥L (S c)) = -, ⊥L c
cut o@(∨R₁ a) (⊥L (S c)) = -, ⊥L c
cut o@(∨R₂ a) (⊥L (S c)) = -, ⊥L c
cut o@⊤R (⊥L (S c)) = -, ⊥L c
-- right rule + left rule
-- the cut proposition is used
cut o@(∧R a b) (∧L₁ Z d) = do x ← cut (wk o) (exch d)
cut a x
cut o@(∧R a b) (∧L₂ Z d) = do x ← cut (wk o) (exch d)
cut b x
cut o@(⊃R a) (⊃L Z d e) = do x ← cut o d
y ← cut x a
z ← cut (wk o) (exch e)
cut y z
cut o@(∨R₁ a) (∨L Z d e) = do x ← cut (∨R₁ (wk a)) (exch d)
cut a x
cut o@(∨R₂ a) (∨L Z d e) = do x ← cut (∨R₂ (wk a)) (exch e)
cut a x
-- sequent calculus is consistent in that `⊥ is not deducible
sc-consistent : · ⇒ `⊥ ^ m → ⊥
sc-consistent (∧L₁ () _)
sc-consistent (∧L₂ () _)
sc-consistent (⊃L () _ _)
sc-consistent (∨L () _ _)
sc-consistent (⊥L ())
-- every natural deduction derivation is a sequent calculus derivation
nd→sc : Γ ⊢ C → ∃[ q ] Γ ⇒ C ^ q
nd→sc (ass x) = id x
nd→sc (∧I x y) = do a ← nd→sc x
b ← nd→sc y
-, ∧R a b
nd→sc (∧E₁ x) = do a ← nd→sc x
b ← id Z
cut a (∧L₁ Z b)
nd→sc (∧E₂ x) = do a ← nd→sc x
b ← id Z
cut a (∧L₂ Z b)
nd→sc (⊃I x) = do a ← nd→sc x
-, ⊃R a
nd→sc (⊃E x y) = do a ← nd→sc x
b ← nd→sc y
c ← id Z
cut a (⊃L Z (wk b) c)
nd→sc (∨I₁ x) = do a ← nd→sc x
-, ∨R₁ a
nd→sc (∨I₂ x) = do a ← nd→sc x
-, ∨R₂ a
nd→sc (∨E x y z) = do a ← nd→sc x
b ← nd→sc y
c ← nd→sc z
cut a (∨L Z (wk′ b) (wk′ c))
nd→sc ⊤I = -, ⊤R
nd→sc (⊥E x) = do a ← nd→sc x
cut a (⊥L Z)
-- every sequent calculus derivation is a natural deduction derivation
sc→nd : Γ ⇒ C ^ m → Γ ⊢ C
sc→nd (idᵖ x) = ass x
sc→nd (∧R x y) = ∧I (sc→nd x) (sc→nd y)
sc→nd (∧L₁ x y) = ⊃E (⊃I (sc→nd y)) (∧E₁ (ass x))
sc→nd (∧L₂ x y) = ⊃E (⊃I (sc→nd y)) (∧E₂ (ass x))
sc→nd (⊃R x) = ⊃I (sc→nd x)
sc→nd (⊃L x y z) = ⊃E (⊃I (sc→nd z)) (⊃E (ass x) (sc→nd y))
sc→nd (∨R₁ x) = ∨I₁ (sc→nd x)
sc→nd (∨R₂ x) = ∨I₂ (sc→nd x)
sc→nd (∨L x y z) = ∨E (ass x) (sc→nd y) (sc→nd z)
sc→nd ⊤R = ⊤I
sc→nd (⊥L x) = ⊥E (ass x)
-- natural deduction is also consistent in that `⊥ is not deducible
nd-consistent : · ⊢ `⊥ → ⊥
nd-consistent x = let ⟨ _ , a ⟩ = nd→sc x
in sc-consistent a
private
-- examples
ex₀ : ∃[ q ] · ⇒ A `⊃ B `⊃ A `∧ B ^ q
ex₀ = do x ← id (S Z)
y ← id Z
-, ⊃R (⊃R (∧R x y))
ex₁ : ∃[ q ] · ⇒ (A `⊃ B `∧ C) `⊃ (A `⊃ B) `∧ (A `⊃ C) ^ q
ex₁ = do x ← id Z
y ← id Z
z ← id Z
w ← id Z
-, ⊃R (∧R (⊃R (⊃L (S Z) x (∧L₁ Z y)))
(⊃R (⊃L (S Z) z (∧L₂ Z w))))
ex₂ : ∃[ q ] · ⇒ A `⊃ A ^ q
ex₂ = do x ← id Z
-, ⊃R x
-- unlike natural deduction, there couldn't be any other proof of A ⊃ A